\(\def\N{\mathbb{N}}\) \(\def\Z{\mathbb{Z}}\) \(\def\Q{\mathbb{Q}}\) \(\def\R{\mathbb{R}}\) \(\def\C{\mathbb{C}}\) \(\def\H{\mathbb{H}}\) \(\def\6{\partial}\) \(\DeclareMathOperator\Res{Res}\) \(\DeclareMathOperator\M{M}\) \(\DeclareMathOperator\ord{ord}\) \(\DeclareMathOperator\const{const}\) \(\DeclareMathOperator{\arccosh}{arccosh}\) \(\DeclareMathOperator{\arcsinh}{arcsinh}\) \(\DeclareMathOperator\id{id}\) \(\DeclareMathOperator\rk{rk}\) \(\DeclareMathOperator\tr{tr}\) \(\def\pt{\mathrm{pt}}\) \(\DeclareMathOperator\colim{colim}\) \(\DeclareMathOperator\Hom{Hom}\) \(\DeclareMathOperator\End{End}\) \(\DeclareMathOperator\Aut{Aut}\) \(\let\Im\relax\DeclareMathOperator\Im{Im}\) \(\let\Re\relax\DeclareMathOperator\Re{Re}\) \(\DeclareMathOperator\Ker{Ker}\) \(\DeclareMathOperator\Coker{Coker}\) \(\DeclareMathOperator\Map{Map}\) \(\def\GL{\mathrm{GL}}\) \(\def\SL{\mathrm{SL}}\) \(\def\O{\mathrm{O}}\) \(\def\SO{\mathrm{SO}}\) \(\def\Spin{\mathrm{Spin}}\) \(\def\U{\mathrm{U}}\) \(\def\SU{\mathrm{SU}}\) \(\def\g{{\mathfrak g}}\) \(\def\h{{\mathfrak h}}\) \(\def\gl{{\mathfrak{gl}}}\) \(\def\sl{{\mathfrak{sl}}}\) \(\def\sp{{\mathfrak{sp}}}\) \(\def\so{{\mathfrak{so}}}\) \(\def\spin{{\mathfrak{spin}}}\) \(\def\u{{\mathfrak u}}\) \(\def\su{{\mathfrak{su}}}\) \(\def\cA{\mathcal{A}}\) \(\def\cB{\mathcal{B}}\) \(\def\cC{\mathcal{C}}\) \(\def\cD{\mathcal{D}}\) \(\def\cE{\mathcal{E}}\) \(\def\cF{\mathcal{F}}\) \(\def\cG{\mathcal{G}}\) \(\def\cH{\mathcal{H}}\) \(\def\cI{\mathcal{I}}\) \(\def\cJ{\mathcal{J}}\) \(\def\cK{\mathcal{K}}\) \(\def\cL{\mathcal{L}}\) \(\def\cM{\mathcal{M}}\) \(\def\cN{\mathcal{N}}\) \(\def\cO{\mathcal{O}}\) \(\def\cP{\mathcal{P}}\) \(\def\cQ{\mathcal{Q}}\) \(\def\cR{\mathcal{R}}\) \(\def\cS{\mathcal{S}}\) \(\def\cT{\mathcal{T}}\) \(\def\cU{\mathcal{U}}\) \(\def\cV{\mathcal{V}}\) \(\def\cW{\mathcal{W}}\) \(\def\cX{\mathcal{X}}\) \(\def\cY{\mathcal{Y}}\) \(\def\cZ{\mathcal{Z}}\) \(\def\al{\alpha}\) \(\def\be{\beta}\) \(\def\ga{\gamma}\) \(\def\de{\delta}\) \(\def\ep{\epsilon}\) \(\def\ze{\zeta}\) \(\def\th{\theta}\) \(\def\io{\iota}\) \(\def\ka{\kappa}\) \(\def\la{\lambda}\) \(\def\si{\sigma}\) \(\def\up{\upsilon}\) \(\def\vp{\varphi}\) \(\def\om{\omega}\) \(\def\De{\Delta}\) \(\def\Ka{{\rm K}}\) \(\def\La{\Lambda}\) \(\def\Om{\Omega}\) \(\def\Ga{\Gamma}\) \(\def\Si{\Sigma}\) \(\def\Th{\Theta}\) \(\def\Up{\Upsilon}\) \(\def\Chi{{\rm X}}\) \(\def\Tau{{T}}\) \(\def\Nu{{\rm N}}\) \(\def\op{\oplus}\) \(\def\ot{\otimes}\) \(\def\t{\times}\) \(\def\bt{\boxtimes}\) \(\def\bu{\bullet}\) \(\def\iy{\infty}\) \(\def\longra{\longrightarrow}\) \(\def\an#1{\langle #1 \rangle}\) \(\def\ban#1{\bigl\langle #1 \bigr\rangle}\) \(\def\llbracket{{\normalsize\unicode{x27E6}}} \def\rrbracket{{\normalsize\unicode{x27E7}}} \) \(\def\lb{\llbracket}\) \(\def\rb{\rrbracket}\) \(\def\ul{\underline}\) \(\def\ol{\overline}\)

10  Residue theorem

Definition 10.1 Let \(z_0\) be an isolated singularity of a holomorphic function \(f\colon U\to\C.\) The residue \(\Res_{z_0}(f)\) is the residue of the Laurent series Equation 9.7. In other words, \[f(z)=\sum_{n=-\iy}^\iy a_n(z-z_0)^n\implies \Res_{z_0}(f)=a_{-1}.\]

Proposition 10.1 Let \(z_0\) be an isolated singularity of both the holomorphic functions \(f,g\colon U\to\C.\)

  1. \(\Res_{z_0}(f+g)=\Res_{z_0}(f)+\Res_{z_0}(g)\)
  2. \(\Res_{z_0}(\la f)=\la\Res_{z_0}(f)\) for all \(\la\in\C\)
  3. If \(z_0\) is a pole of order \(m,\) then \(\Res_{z_0}(f)=\frac{1}{(m-1)!}\lim_{z\to z_0}h^{(m-1)}(z)\) for \(h(z)=(z-z_0)^mf(z).\)

Proof.

Covered in lectures. Check back once the chapter is concluded.






Example 10.1  

Covered in lectures. Check back once the chapter is concluded.




Example 10.2  

Covered in lectures. Check back once the chapter is concluded.
















Definition 10.2 Let \(\ga\colon[a,b]\to\C\) be a closed piecewise C1 curve and let \(z_0\in\C\setminus\ga([a,b]).\) The winding number of \(\ga\) around \(z_0\) is\[W_{z_0}(\ga)=\frac{1}{2\pi i}\int_\ga\frac{dz}{z-z_0}.\]

Example 10.3  

Covered in lectures. Check back once the chapter is concluded.






Pick a subdivision \(a=t_0<t_1<\cdots<t_n=b\) such that \(\ga|_{[t_{k-1},t_k]}\) is continuously differentiable. Using branches of the logarithm, we can write \(\ga(t)=z_0+r(t)e^{i\th(t)}\) for \(r(t)=|\ga(t)-z_0|\) and a continuous function \(\th\colon[a,b]\to\R\) with \(\th|_{[t_{k-1},t_k]}\) continuously differentiable. With this notation, the following holds.

Proposition 10.2  

  1. We have \(W_{z_0}(\ga)=\frac{\th(b)-\th(a)}{2\pi},\) which is an integer.
  2. For closed curves \(\ga_0,\) \(\ga_1\) in \(\C\setminus\{z_0\}\) we have \[W_{z_0}(\ga_0)=W_{z_0}(\ga_1)\iff\text{$\ga_0,$ $\ga_1$ are homotopic in $\C\setminus\{z_0\}.$}\]

Proof.

Covered in lectures. Check back once the chapter is concluded.














Theorem 10.1 (Residue theorem) Let \(U\subset\C\) be an open set, \(\ga\colon[a,b]\to U\) a null-homotopic piecewise C1 loop in \(U,\) and \(z_1,\ldots, z_n\in U\setminus\ga([a,b])\) points not on the loop \(\ga.\) If \(f\) is a holomorphic function on \(U\setminus\{z_1,\ldots, z_n\},\) then

\[\frac{1}{2\pi i}\int_\ga f(z)dz=\sum_{k=1}^n W_{z_k}(\ga)\Res_{z_k}(f).\]

Proof.

Covered in lectures. Check back once the chapter is concluded.

















10.1 Exercises

Exercise 10.1

Determine the winding numbers \(W_{z_0}(\ga)\) in the following cases.

  1. \(\ga\colon[0,2\pi]\to\C,\) \(\ga(t)=e^{it},\) for \(z_0=0,2.\)
  2. \(\ga\colon[2\pi,10\pi]\to\C,\) \(\ga(t)=\begin{cases} te^{it} & \text{if $t\in[2\pi, 6\pi],$}\\ 12\pi - t & \text{if $t\in[6\pi,10\pi],$} \end{cases}\) for \(z_0=0, 20.\)
  3. \(\ga\colon[0,2\pi]\to\C,\) \(\ga(t)=(2+\cos(t))e^{2it},\) for \(z_0=0,4.\)
Exercise 10.2

Use the residue theorem to compute \[\int_{\partial D_r(0)}\frac{(z+2)^2}{z(z-1)^2}dz\]

for all \(0<r\neq1.\)

Hint: Consider the cases \(r>1\) and \(r<1\) separately.

Exercise 10.3

Apply the residue theorem to the contour \(\partial D_1(0)\) and a suitable holomorphic function \(f(z)\) to compute the integral\[\int_0^{2\pi}\frac{dx}{2+\cos(x)}.\]

Exercise 10.4

Compute the integral \[\int_{0}^{+\iy}\frac{\cos(x)}{\sqrt{x}}dx\] by considering \(f(z)=\frac{e^{iz}}{\sqrt{z}}\) and the following contour \(\ga_{r,R}\) for \(r\to 0,\) \(R\to+\iy.\)

Hint: Recall that \(\int_0^{\iy}e^{-x^2}dx=\sqrt{\pi}/2.\)

Exercise 10.5

Compute the integral \[\int_{-\iy}^{+\iy}\frac{dx}{1+x^6}\]

by applying the residue theorem to \(f(z)=\frac{1}{1+z^6}\) and the following upper semi-circular contour \(\ga_R\) for \(R\to+\iy.\)